In a saucepan, over medium-low heat, combine 1/2 cup of the oil and the shallots. Simmer until the shallots are light golden brown, about 12 minutes. Transfer to a bowl and stir in the capers, vinegar, 1/4 teaspoon salt, and 1/4 teaspoon pepper; set aside.
Heat 1 tablespoon of the oil in a large skillet over medium-high heat.
Season the fish with 1/4 teaspoon salt and 1/4 teaspoon pepper and cook until opaque, about 4 minutes per side. Divide among plates.
Wipe out the skillet and heat the remaining oil over medium heat. Add the spinach, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Cook, stirring, until wilted, 2 to 3 minutes. Serve with the striped bass, drizzling the vinaigrette over the top.